Аннотация:We study linear transient dynamics in a thin Keplerian disc, employing a method based on a variational formulation of the optimization problem. It is shown that in a shearing sheet approximation due to a prominent excitation of density waves by vortices, the most rapidly growing shearing harmonic has an azimuthal wavelength λy, of the order of the disc thickness H, and its initial shape is always nearly identical to a vortex having the same potential vorticity. Also, in the limit λy ≫ H, the optimal growth G ∝ (Ω/κ)4, where Ω and κ denote local rotational and epicyclic frequencies, respectively. This suggests that the transient growth of large-scale vortices can be much stronger in areas with non-Keplerian rotation (e.g. in the inner parts of relativistic discs around black holes). We estimate that if the disc is already in a turbulent state with effective viscosity given by the Shakura parameter α < 1, the considered large-scale vortices with wavelengths H/α > λy > H have the most favourable conditions to be transiently amplified before they are damped. At the same time, turbulence is a natural source of the potential vorticity for this transient activity. We extend our study to a global spatial scale, showing that global perturbations with azimuthal wavelengths more than an order of magnitude greater than the disc thickness are still able to attain the growth of dozens of times in a few Keplerian periods at the inner boundary of the disc.

Zhuravlev V. V., Razdoburdin D. N. A study of the transient dynamics of perturbations in keplerian discs using a variational approach // Monthly Notices of the Royal Astronomical Society. — 2014. — Vol. 442. — P. 870–890. We study linear transient dynamics in a thin Keplerian disc, employing a method based on a variational formulation of the optimization problem. It is shown that in a shearing sheet approximation due to a prominent excitation of density waves by vortices, the most rapidly growing shearing harmonic has an azimuthal wavelength λy, of the order of the disc thickness H, and its initial shape is always nearly identical to a vortex having the same potential vorticity. Also, in the limit λy ≫ H, the optimal growth G ∝ (Ω/κ)4, where Ω and κ denote local rotational and epicyclic frequencies, respectively. This suggests that the transient growth of large-scale vortices can be much stronger in areas with non-Keplerian rotation (e.g. in the inner parts of relativistic discs around black holes). We estimate that if the disc is already in a turbulent state with effective viscosity given by the Shakura parameter α < 1, the considered large-scale vortices with wavelengths H/α > λy > H have the most favourable conditions to be transiently amplified before they are damped. At the same time, turbulence is a natural source of the potential vorticity for this transient activity. We extend our study to a global spatial scale, showing that global perturbations with azimuthal wavelengths more than an order of magnitude greater than the disc thickness are still able to attain the growth of dozens of times in a few Keplerian periods at the inner boundary of the disc. [ DOI ]